[2512.19196] Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equations

Physics > Computational Physics arXiv:2512.19196 (physics) [Submitted on 22 Dec 2025 (v1), last revised 24 Mar 2026 (this version, v3)] Title:Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equations Authors:Xiaolong Wu, Qifeng Liao View a PDF of the paper titled Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck Equations, by Xiaolong Wu and Qifeng Liao View PDF HTML (experimental) Abstract:Solving high-dimensional Fokker-Planck (FP) equations is a challenge in computational physics and stochastic dynamics, due to the curse of dimensionality (CoD) and unbounded domains. Existing deep learning approaches, such as Physics-Informed Neural Networks, face computational challenges as dimensionality increases, driven by the $O(d^2)$ complexity of automatic differentiation for second-order derivatives. While recent probability flow approaches bypass this by learning score functions or matching velocity fields, they often involve serial operations or depend on sampling efficiency in complex distributions. To address these issues, we propose the Adaptive Probability Flow Residual Minimization (A-PFRM) method. The second-order FP equation is reformulated as an equivalent first-order deterministic Probability Flow ODE (PF-ODE) constraint, which avoids explicit Hessian computation. Unlike score matching or velocity matching, A-PFRM solves FP equations by minimizing the residual of the continuity equation induced by the...